In physics, quantitative models are developed on the basis of measurements. They need to be tested by making more measurements. Measurements are made in standard increments, called units. Without units a
measurement is meaningless.

Some quantities are specified completely by a magnitude, which is to say by a
number and the appropriate units.

A real number by itself is called a
scalar, and such quantities are called **scalar quantities**. Symbols that denote these scalar quantities are
normal letters.

temperature (T) T = 10 ^{o}C

time interval (t) t = 5 s

mass (m) m = 3 kg

Other quantities are specified by a magnitude and a direction. By
direction, we mean a direction in space.

Such quantities are called
**vector quantities**. Symbols that denote
these vector quantities are bold letters, or normal letters with arrows drawn
above.

displacement (**d**) **d** = 10 m north

velocity (**v**)
**v** = 3 m/s eastward

force (**F**)
**F** = 9 N up

To uniquely specify vector quantities we need a
**reference point and reference lines**, i.e. we need a
**coordinate system**.

The **Cartesian coordinate
system** is the most commonly used coordinate system. In
two dimensions, this system consists of a pair of lines on a flat
surface or plane, that intersect at right angles. The lines are
called **axes** and the point at which
they intersect is called the **origin**.
The axes are usually drawn horizontally and vertically and are referred
to as the x- and y-axes, respectively.

A point in the plane with
coordinates (a, b) is a units to the right of the y axis and b units up
from the x axis if a and b are positive numbers.

If a and b are
both negative numbers, the point is a units to the left of the y axis
and b units down from the x axis.

In the figure on the right point
P_{1} has coordinates (3, 4), and point P_{2} has
coordinates (-1, -3).

In three-dimensional Cartesian coordinates, the z axis is added so that
there are three axes all perpendicular to each other.

In the **polar coordinate** **system**,
each point in the plane is assigned coordinates (r, φ) with respect to a
fixed line in the plane called the **axis****
**and a point on that line called the **pole**.
For a point in the plane, the r-coordinate is the distance from the
point to the pole, and the φ-coordinate is the counterclockwise angle
between the axis and a line joining the origin to the point.

The
r-coordinate is always positive and the range of φ is from 0 to 2π (360^{o}).
To be able to transform from Cartesian to polar coordinates and vice
versa, we let the axis of the polar coordinate system coincide with the
x-axis of the Cartesian coordinate system and the pole coincide with the
origin.

In the figure on the right he Point P_{1} has
polar coordinates (r_{1}, φ_{1}) = (5, 53.1^{o}),
and the point P_{2} has polar coordinates (r_{2}, φ_{2})
= (3.16, 251.6^{o} ).

The transformation equations are

x = r cosφ, y = r sinφ.

r = (x^{2 }+ y^{2})^{½}, φ =
tan^{-1}(y/x).

Cylindrical coordinates and spherical coordinates are
two different extensions of polar coordinates to three dimensions.

Once we have picked a coordinate system, a physical
vector quantity in two dimensions can be represented by a pair of
numbers.

If we pick Cartesian coordinates, there numbers are the vector's projections on the axes of
the Cartesian coordinate
system.

The vector
**a** in the figure on the right has x-component
a_{1} and y-component a_{2}.

Its length or magnitude is a = (a_{1}^{2} + a_{2}^{2})^{½}.
This follows from the
Pythagorean theorem.

The polar angle of **a**, i.e. the angle **a** makes with the
x-axis is φ.

A vector **A** lies in the xy-plane.

(a) For what orientations of
**
A** will both of its rectangular components be negative?

(b) For what
orientation will its components have opposite signs?

Solution:

(a)
In a plane (in two dimensions), we can specify the direction of a vector by
setting up a coordinate system and giving the angle φ the vector makes with
the positive x-axis. A positive φ is an angle measured counterclockwise
from the positive x-axis.

The x-component of A is A_{x }= A cosφ
and the y-component of A is A_{y }= A sinφ. (A is the magnitude
of the vector.) For A_{x} and A_{y} to be negative, we
need cosφ and sinφ to be negative. φ must lie between 180 and 270
degrees.

(b) For A_{x} and A_{y} to have opposite signs, φ must lie
between 90 and 180 degrees or between 270 and 360 degrees.

To add physical vectors, they
have to have the same units. To find the sum of two physical
vector quantities with the same units **algebraically**, we add the
x, y, and z-components of the individual vectors.

Let vector **v**_{1} have components (3, 4) and vector
**
v**_{2} have components (2, -3).

Let **v** =
**
v**_{1} + **v**_{2} be the sum of the two vectors.

Then the components of
**v** are (3+2, 4+(-3) = (5, 1).

The magnitude of the vector **v** is v = (25 + 1)^{½} = 5.1,

and the angle **v** makes with the x-axis is φ = tan^{-1}(1/5)
= 0.197 rad = 11.3^{o}.

To subtract vector
**v**_{2} from vector **v**_{1}
we subtract the components of vector **v**_{2} from the
components of vector **v**_{1}.

Let vector **v**_{1} have components (3, 4) and vector
**
v**_{2} have components (2, -3).

Let **v** =
**
v**_{1} - **v**_{2} be the difference of the two
vectors.

Then the components of **v** are (3-2, 4-(-3) = (1,
7).

The magnitude of the vector **v** is v = (1 + 49)^{½} = 7.1

and the angle **v** makes with the x-axis is φ = tan^{-1}(7/1)
= 1.429 rad = 81.9^{o}.

The graphical representation of a vector quantity is a directed line
segment. To find the sum of two physical vector quantities with
the same units **graphically** we line up the arrows, tail to tip.
The sum is the arrow drawn from the tail of the first vector to the tip
of the last vector. To subtract a vector **v**_{2} from
a vector **v**_{1} we we invert vector **v**_{2}
and add it to vector **v**_{1}.

Let **A** be an arbitrary vector. The vector -**A** has the
same length, and points in exactly the opposite direction.
Subtracting the vector **A** from another vector means adding the
vector -**A** to the other vector.

The displacement vector

Let **d** represent the displacement vector from point
**A** with
coordinates (x_{1}, y_{1}) = (-4, -1) to point **B**
with coordinates (x_{2}, y_{2}) = (3, 4).

The displacement vector is the difference between the position vectors
**A** and **B**, **d** =
**B** - **A**.

Its components
are

d_{x }= (x_{2} - x_{1}) = 3 - (-4) = 7, d_{y
}= (y_{2} - y_{1}) = 4 - (-1) = 5.

The displacement vector **d** has magnitude d = (49 + 25)^{½}
= 8.6.

The angle **d** makes with the x-axis is φ = tan^{-1}(5/7) =
0.62 rad = 35.5^{o}.

Link: Lesson 1: Vectors - Fundamentals and Operations

Please study this material from "the Physics Classroom".